It could be that the socket has been wired incorrectly ie, live and neutral reversed. Another possibility is that the switch has become stuck in the "on" position.
Your own suggestion of replacing the socket is a good one.
Probably just a duff switch or some other reason why the circuit is shorting inside the socket (such as water ingress).
It's fairly common. I had a similar problem with a kitchen worktop socket just a few weeks ago. (I suspect that many years of being close to a steaming kettle hadn't helped!). £1.14 for a new double socket from Screwfix was all that was needed to fix the problem.
The connections on the back of the new socket are marked. You should have two cables going to it. Stripey wires go to the connection marked earth. Bizarrely the mud coloured ones connect to live. Blues to neutral.
You may have a stressed lead to the d/washer Hopkirk. Inspect all the lead for cracks or kinks particularly where it goes into the plug and the machine. When you rewire a new plug cut off a few inches of the lead.
Hoppy......you have a Neutral to earth fault within the flex or appliance.The socket feeding the appliance is a single pole switching type meaning ...the Neutral is still connected even when the switch is off.
Hi sparks, I did not want to make a definitive statement but was hoping you would show. I am with you all day on that (that is why I said check the supply lead). Hope you are good.